The Yamnistsky-Levin algorithm (1982) is a variation of the ellipsoid
method that uses a simplex instead of an ellipsoid. Unlike the
ellipsoid method, it does not require square root computations, and it
still runs in weakly polynomial time. However, its running time is
still not practical. We introduce a variation of this method that
examines a subset of cutting planes, and uses the Multiplicative
Weights framework developed by Plotkin, Shmoys and Tardos for packing
linear programs. We will show how our algorithm improves the original
YL algorithm for a simplification of the problem, and what questions
remain open to give a complete proof of convergence.

Romanos Malikiosis

24.01.18

16:30

MA 406

Fuglede's spectral set conjecture on cyclic groups

Abstract:

Fuglede's conjecture (1974) states that a bounded measurable subset in \(\mathbb{R}^d\) accepts an orthogonal basis of exponential functions (i.e. it is spectral) if and only if it tiles the space with a discrete set of translations. This conjecture turned out to be false by Tao's counterexample in 2003. Using Tao's ideas, counterexamples in finite Abelian groups such as \(\mathbb{Z}_N^d\) can be lifted to counterexamples in \(\mathbb{R}^d\), thus shifting the interest on this conjecture to this setting in recent years. This has been successful for \(d\geq3\), but the conjecture is still open for \(d=1,2\). Some recent results in the cyclic group setting will be presented in this talk, which are connected to the work of Coven-Meyerowitz and Laba on tiling subsets of \(\mathbb{Z}\), as well as the structure of vanishing sums of roots of unity. This is joint work with Mihalis Kolountzakis & work in progress.

Linda Kleist

20.12.17

16:30

MA 621

Rainbow cycles in flip graphs

Abstract:

The vertices of flip graph are combinatorial or geometric objects, and its edges link two of these objects whenever they can be obtained from one another by an elementary operation called a flip. With each edge, we associate the type of a flip (thought of as a color) and seek for r-rainbow cycles, i.e., cycles where every flip type occurs exactly r times.For instance, the flip graph of triangulations has as vertices all triangulations of a convex n-gon, and an edge between any two triangulations that differ in exactly one edge. An r-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly r times.In this talk we investigate the existence of rainbow cycles in some of
the following flip graphs of triangulations of a convex n-gon, plane
trees on a set of n points, non-crossing perfect matchings on points in
convex position, permutations of [n], and the flip graph of k-element
subsets of [n].The talk is based on joint work with Stefan Felnser, Torsten Mütze and
Leon Sering.

Franz Schuster (TU Wien)

06.12.17

16:30

MA 406

Affine vs. Euclidean Sobolev inequalities

Abstract:

In this talk we explain how every even, zonal measure on the Euclidean unit sphere gives rise to a sharp Sobolev inequality for functions of bounded variation which directly implies the classical Euclidean Sobolev inequality. The strongest member of this large family of inequalities is shown to be the only affine invariant one among them – the affine Zhang-Sobolev inequality. We discuss in some detail the geometry behind these analytic inequalities and also relate our new Sobolev inequalities to the sharp Gromov-Gagliardo-Nirenberg Sobolev inequality for general norms and discuss further improvements of special cases.

Gennadiy Averkov (OvGU Magdeburg)

29.11.17

16:30

MA 406

Latest news on Hensley's conjecture

Abstract:

The talk is about the maximum volume of d-dimensional lattice polytopes with a fixed positive number of interior lattice points. Several researchers conjectured that the maximum is attained on a nice family of polytopes arising from the so-called Sylvester sequence. I will report on the recent progress on Hensley's conjecture. This is joint work with Benjamin Nill and Jan Krümpelmann.

Moment-sum-of-squares hierarchies of semidefinite programs can be used
to approximate the volume of a given compact basic semialgebraic set K.
The idea consists of approximating from above the indicator function of
K with a sequence of polynomials of increasing degree d, so that the
integrals of these polynomials generate a convergence sequence of upper
bounds on the volume of K. In this talk we show how we could derive an
asymptotic rate of convergence for this approximation scheme. Joint work
with Milan Korda, see arXiv:1609.02762.

Henning Seidler

15.11.17

16:30

MA621

Circuit Polynomials for Simplex Newton Polytopes

Abstract:

Finding the minimum of a multivariate real polynomial is a well-known
hard problem with various applications. We present an implementation to approximate
such lower bounds via sums of non-negative circuit polynomials (SONCs). We provide a
test-suite, where we compare our approach, using different solvers, with several solvers
for sums os squares (SOS), including sostools and gloptipoly. It turns out that the
circuit polynomials yield bounds competitive to SOS in several cases, but using much
less time and memory.

In this talk we present the SoS hierarchy in the
context of 0/1 optimization along with some of our recent
integrality gap/lower bound results. We introduce the
relevant concepts and give a brief derivation of the
hierarchy. As for our results, we characterize the
possible gap instances on the round n-1 of the hierarchy.
Then we show an unbounded integrality gap after
nonconstant number of rounds for a problem that admits a
polynomial time algorithm. Finally we present a
simplification of the positive semidefiniteness
constraints when a high degree of symmetry can be assumed,
and some applications to integrality gaps and lower
Bounds.

Daryl Cooper

13.09.17

16:30

MA621

Limits of Algebraic varieties: towards a continuous Nullstellansatz

Abstract:

The map that sends a polynomial ideal to an algebraic
variety in complex affine space is not continuous with respect to the
obvious
topology on the domain. However it is continuous on the closure of
the set of ideals generated by polynomials of degree at most d.
Moreover the closure of this subspace is compact.
This is work in progress, and is joint with Ricky Demer.

Dr. Iskander Aliev (Cardiff, Wales)

05.09.17

16:30

MA 406

Sparse solutions to the systems of linear Diophantine equations

Abstract:

We present structural results on sparse nonnegative integer solutions to underdetermined systems of linear equations.Our proofs are algebraic and number theoretic in nature and make use of Siegel's Lemma and classical tools from the Geometry of Numbers.These results have some interesting consequences in discrete optimisation.

Marta Panizzut

30.08.17

16:30

MA 621

The hunt for K3 polytopes

Abstract:

The connected components of the complement of a tropical hypersurface
are called regions, and they are convex polyhedra. A smooth tropical quartic
surface of genus one has exaclty one bounded region. We say that a 3-
dimensional polytope is a K3 polytope if it arises as the bounded region of
a smooth tropical quartic surface.
In a joint project with Gabriele Balletti, we are investigating properties
of K3 polytopes. In this talk I will report on the progress of this work.

Timo de Wolff

23.08.17

16:30

MA 621

Intersections of Amoebas

Abstract:

Georg Loho

16.08.17

16:30

MA 621

Monomial Tropical Cones for Multicriteria Optimization

Abstract:

We introduce a special class of tropical cones which we call 'monomial
tropical cones'.
They arise as a crucial tool in the description of discrete
multicriteria optimization problems. We employ them to derive an
algorithm for computing all nondominated points.
Furthermore, monomial tropical cones are intimately related with
monomial ideals. We sketch directions for further work.
This is based on joint work with Michael Joswig.

Johannes Hofscheier

19.07.17

16:30

MA 621

A Combinatorial Smoothness Criterion for Spherical Varieties

Abstract:

In this talk we will present algorithms to check smoothness for the class of spherical varieties which contains those of toric varieties, flag varieties and symmetric varieties, and which forms a remarkable class of algebraic varieties with an action by an algebraic group having an open dense orbit. In the toric case there is a well-known simple combinatorial smoothness criterion whereas in the spherical case Brion, Camus and Gagliardi have shown smoothness criteria for spherical varieties which either are rather involved or rely on certain classification results. We suggest a purely combinatorial smoothness criterion by introducing a rational invariant which only depends on the combinatorics of the spherical variety. We have a conjectural inequality this invariant should satisfy where the equality case would imply that the spherical variety has a toric torus action. Our conjecture would also imply the generalized Mukai conjecture for spherical varieties. We complete our talk by summarizing in which cases the above mentioned approach is known to be true. This is joint work with Giuliano Gagliardi.

Sinai Robins (University of São Paulo)

06.07.17

16:30

MA 406

The Ehrhart quasipolynomials of polytopes that come from trivalent graphs, and a discrete version of Hilbert's 3rd problem for the unimodular group

Abstract:

A graph all of whose nodes have degree \(1\) or \(3\) is called a \(\{1,3\}\)-graph. Liu and Osserman associated to each \(\{1, 3\}\)-graph a polytope. They studied these polytopes and their Ehrhart quasi-polynomials. We prove a conjecture of Liu and Osserman stating that the associated polytopes of all connected \(\{1,3\}\)-graphs with the same number of nodes and edges have the same Ehrhart quasi-polynomial. We also present some structural properties of these polytopes and we show a relation between them and the well-known metric polytopes. This study is related to a question of Hasse and McAllister on a discrete version of Hilbert's 3rd problem for the unimodular group.This is joint work with Cristina Fernandez, José C. de Pina, and Jorge Luis Ramírez Alfonsín.

Matthias Lenz

28.06.17

16:30

MA 621

On powers of Plücker coordinates and representability of arithmetic matroids

Abstract:

Given \(k\in \mathbb{R}_{\ge 0}\) and a vector \(v\) of Plücker
coordinates of a point in the real Grassmannian, is the vector obtained by
taking the \(k\)th power of each entry of \(v\) again a vector of
Plücker coordinates? We will show that for \(k\neq 1\), this is true if
and only if the corresponding matroid is regular. Similar results hold over
other fields. We will also describe the subvariety of the Grassmannian that
consists of all the points that define a regular matroid.An arithmetic matroid is a matroid equipped with a function that assigns a
multiplicity (a positive integer) to each subset of the ground set. We will
discuss a discrete version of the problem described above that deals with the
representability of the arithmetic matroid obtained by taking the \(k\)th
power of the multiplicity function. This setting is motivated by some
combinatorial questions on CW complexes.

Jean-Philippe Labbé / Michael Joswig

21.06.17

16:30

MA 621

polymake/sage

Abstract:

Jean-Philippe will give an introduction to the new polymake interface in sage. Michael will report basic design principles underlying polymake. This is followed by a common coding session and beer.

Héctor Andrade Loarca

14.06.17

16:30

MA 621

Fast Multidimensional Signal Processing with Shearlab.jl.

Abstract:

We live in the age of data, huge amount of Data its been generated and acquired everyday in different forms and flavors. One would like to have a technique to store and process optimally this data; this is possible since the relevant information in almost all data found in typical applications is sparse due the high correlation of its elements. The challenge will lie in the search of appropriate dictionary that can represent optimally this information. Along the history very talented scientist have proposed different signal transforms based on certain representation dictionaries, starting with the Fourier Transform and Short Time Fourier Transform; in 1980s the wavelet transform was proposed representing a breakthrough in one dimensional signal representation, with the ability to not just optimally represent the data but also capture certain singularities, regularities and other features; the flaw of the wavelets transforms is its lack of directional sensitivity which does not aloud it to represent optimally multidimensional singularities like curves that are predominant in general multidimensional signals (for instance images and videos). The Shearlet transform is a product of almost 10 years of research on sparsifying transform with directional sensitivity that attains the optimal best N-term approximation error, it was proposed by G. Kutyniok, D. Labate and K Guo in 2015 and since then it's been known to be the best of its type.
In this talk I will explain the upsides and downsides of the already mentioned transforms, why the shearlet transform is great solution for anisotropic representation of multidimensional signals, how can we implement wavelet and shearlet transform in Julia, and I will show you some applications and demos of the Shearlet Transform Library that I implemented with Professor Gitta Kutyniok based on the matlab version developed by her group two years ago; I will also explain why Julia is the best option to implement algorithms in Signal and Image Processing and show some benchmarks that show its great performance.

In this paper, we address the following algebraic generalization of the bipartite stable set problem. We are given a block-structured matrix (partitioned matrix) \(A=(A_{\alpha\beta})\), where \(A_{\alpha\beta}\) is an \(m_{\alpha}\) by \(n_{\beta}\) matrix over field F for \(\alpha=1,2,\ldots,\mu\) and \(\beta=1,2,\ldots,\nu\). The maximum vanishing subspace problem (MVSP) is to maximize \(\sum_{\alpha}\mbox{dim}X_{\alpha}+\sum_{\beta}\mbox{dim}X_{\beta}\) over vector subspaces \(X_{\alpha} \subseteq F^{m_{\alpha}}\) for \(\alpha=1,2,\ldots,\mu\) and \(Y_{\beta}\subseteq F^{n_{\beta}}\) for \(\beta=1,2,\ldots,\nu\) such that each \(A_{\alpha\beta}\) vanishes on \(X_{\alpha}\times Y_{\beta}\) when \(A_{\alpha\beta}\) is viewed as a bilinear form \(F^{m_{\alpha}}\times F^{n_{\beta}}\rightarrow F\). This problem arises from a study of a canonical block-triangular form of \(A\) by Ito, Iwata, and Murota (1994). We prove that MVSP can be solved in polynomial time. Our proof is a novel combination of submodular optimization on modular lattice and convex optimization on CAT(0)-space. We present implications of this result on block-triangularization of partitioned matrix.

Ted Bisztriczky (University of Calgary)

31.05.17

16:30

MA 406

Erdos-Szekeres type theorems for planar convex sets

Abstract:

A family \(\mathcal{F}\) of sets is in convex position if none of its members is contained in the convex hull of the union of the others. The members of \(\mathcal{F}\) are ovals (compact convex sets) in the plane that have a certain property. An Erdos-Szekeres type theorem concerns the existence, for any integer \(n\geq 3\), of a smallest positive integer \(N(n)\) such that if \(|\mathcal{F}|≥N(n)\) then there are \(n\) ovals of \(\mathcal{F}\) in convex position.In this talk, I survey some known theorems, introduce a new one based upon work with Gabor Fejes Toth and examine the relation between \(N(n)\) and Ramsey numbers of the type \(R_k(n_1,n_2)\).

Dimitrios Dais (University of Crete)

29.05.17

17:00

MA 406

Toric log del Pezzo surfaces with a unique singularity

Abstract:

The toric log del Pezzo surfaces are constructed by means of the so-called LDP-lattice polygons. In the talk it will be explained how one classifies the subclass of surfaces of this kind which have a unique singularity.

Richard Gardner (Western Washington University)

17.05.17

16:30

MA 406

Open problems in Geometric Tomography

Abstract:

This talk will focus on open problems in Geometric Tomography, which aims to retrieve information about a geometric object (such as a convex body, star body, finite set, etc.) from data concerning its intersections with planes or lines and/or projections (i.e., shadows) on planes or lines. The problems, which span nearly a hundred years of mathematics, are diverse. Many have a common thread, however, since they are linked to various integral transforms: the X-ray transform, divergent beam transform, circular Radon transform, cosine transform, or spherical Radon transform.The talk will be illustrated by plenty of pictures.

Michael Joswig

03.05.17

16:30

MA 621

Computing Invariants of Simplicial Manifolds

Abstract:

This is a brief survey on algorithms for computing basic algebraic invariants such as homology, cup products, and intersection forms.

Romanos Malikiosis

26.04.17

16:30

MA 406

Formal duality in finite cyclic groups

Abstract:

Numerical computations by Cohn, Kumar, and Schürmann in energy minimizing periodic configurations of density \(\rho\) and \(1/\rho\) revealed an impressive kind of symmetry, the so–called formal duality. Formal dual subsets of \(\mathbb{R}^n\) satisfy a generalized version of Poisson summation formula, and it is conjectured that the only periodic subsets of \(\mathbb{R}\) of density \(1\) possessing a formal dual subset, are \(\mathbb{Z}\) and \(2\mathbb{Z}\cup(2\mathbb{Z}+\frac12)\).In this talk, we will present recent progress on this conjecture by the speaker, utilizing several tools from different areas of mathematics, such as (a) the field descent method by Bernhard Schmidt, used chiefly on the circulant Hadamard matrix conjecture, (b) the 'polynomial method' by Coven & Meyerowitz used in the characterization of sets tiling \(\mathbb{Z}\) or \(\mathbb{Z}_N\), and (c) basic arithmetic of cyclotomic fields.

Manuel Radons

12.04.17

16:30

MA 621

Piecewise linear methods in nonsmooth optimization

Abstract:

Several techniques in numerical analysis, e.g. Newton's methods and ODE solvers, are based on local linear approximations of a smooth function \(f: \mathbb R^n\rightarrow \mathbb R^n\). If \(f\) is piecewise smooth such approximations may be arbitrarily bad near nondifferentiabilites. Piecewise linear approximations -- or, briefer, piecewise linearizations -- restore the structural correspondence between approximation and underlying function. We give an overview of the generalized numerical methods based on piecewise linearizations and their properties, e.g. convergence results, as well as the techniques used in their investigation, which include degree theory for piecewise affine functions and nonsmooth analysis.

Philipp Jell

05.04.17

16:30

MA 621

Differential forms on Berkovich spaces and their relation to tropical geometry

Abstract:

Takayuki Hibi (Osaka University)

29.03.17

16:30

MA 621

Order Polytopes and Chain Polytopes

Abstract:

Given a finite partially ordered set P, we associate two
poset polytopes, viz, the order polytope O(P) and the chain polytope
C(P). In my talk, after reviewing fundamental materials on O(P) and
C(P), the question when O(P) and C(P) are unimodularly equivalent will
be discussed. Then, in the frame of Gröbner bases, various reflexive
polytopes arising from O(P) and C(P) will be presented. Finally, some
questions and conjectures will be summarized. No special knowledge
will be required to understand my talk.

Lucía López de Medrano (UNAM Cuernavaca)

22.03.17

17:00

MA 621

On the genus of tropical curves

Abstract:

Felipe Rincón (University of Oslo)

22.03.17

16:15

MA 621

Tropical Ideals

Abstract:

Skip Jordan

22.03.17

15:30

MA 621

Parallel Vertex and Facet Enumeration with mplrs

Abstract:

We introduce mplrs, a new parallel vertex/facet enumeration program based on the reverse-search code lrs. The implementation uses MPI and can be used on single machines, clusters and supercomputers. We describe the budgeted parallel tree search implemented in mplrs and compare performance with other sequential and parallel programs for vertex/facet enumeration. In some instances, mplrs achieves almost linear scaling with more than 1000 cores. The approach used in mplrs can be easily applied to various other problems. This is joint work with David Avis.

Alex Fink

22.02.17

16:30

MA 621

The characteristic polynomial two ways

Abstract:

The theory of hyperplane arrangements and matroids derives great utility from the application of algebraic and algebro-geometric methods. The characteristic polynomial often appears in familiar guise in these methods, as an enumerator of the no broken circuit sets. The characteristic polynomial also appears, on the face of it unrelatedly, in the recent magisterial resolution of Rota's conjecture by Huh, Katz, and Adiprasito. In joint work with David Speyer and Alex Woo we explain how these two manifestations are related after all.

Akiyoshi Tsuchiya

08.02.17

16:30

MA 621

Gorenstein simplices and the associated finite abelian groups

Abstract:

A lattice polytope is a convex polytope each of whose vertices has integer coordinates.
It is known that a lattice simplex of dimension \(d\) corresponds to a finite abelian subgroup of \((\mathcal{R}/\mathcal{Z})^{d+1}\).
Conversely, given a finite abelian subgroup of \((\mathcal{R}/\mathcal{Z})^{d+1}\) such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension \(d\).
In this talk, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. Gorenstein polytopes are of interest in combinatorial commutative algebra, mirror symmetry and tropical geometry. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equal \(p\), \(p^2\) and \(pq\), where \(p,q\) are prime integers.

Robert Löwe

25.01.17

16:30

MA 621

Secondary fans of a punctured Riemann surface

Abstract:

The secondary fan of a finite point configuration in \(R^n\) stratifies the space of height functions by the combinatorial types of regular subdivisions.
Now given a punctured Riemann surface, similar techniques yield a polyhedral fan whose cones correspond to horocyclic Delaunay tessellation in the sense of Penner's convex hull construction.
The purpose of the talk is to give an introduction to this topic and the corresponding problems that arise naturally.

Eva Maria Feichtner (Universität Bremen)

18.01.17

16:30

MA 406

A Leray model for Orlik-Solomon algebras

Abstract:

Although hyperplane arrangement complements are rationally formal,
we note that they have non-minimal rational (CDGA) models which are topologically and combinatorially significant. We construct a family of CDGAs which interpolates between the Orlik-Solomon algebra and the cohomology algebras of arrangement compactifications. Our construction is combinatorial and extends to all matroids, regardless of their (complex) realizability.This is joint work with Christin Bibby and Graham Denham.

Benjamin Schröter

04.01.17

16:30

MA 621

Fundamental Polytopes of Metric Trees via Hyperplane Arrangements

Abstract:

In this talk I will present a summary of the recent paper 'Fundamental Polytopes of Metric Trees via Hyperplane Arrangements' (arXiv:1612.05534) of Emanuele Delucchi and Linard Hossly. In this paper they develop a formula for the f-vector of the fundamental polytope of a finite metric space and its dual the Lipschitz polytope.

Georg Loho

04.01.17

17:15

MA 621

Slicing and Dicing Polytopes

Abstract:

I present the recent work 'Slicing and dicing polytopes' (arXiv:1608.05372) by Patrik Norén on cellular resolutions. I introduce diced and sharp polytopes which are necessary for the construction and give a brief introduction to cellular resolutions. The main result, introducing a new class of polyhedral subdivisions which support cellular resolutions, is illustrated along some examples.

A combinatorial theory of linear systems on graphs and metric graphs has been introduced in analogy with the one on algebraic curves. The interplay is given by the Specialization Lemma. Let \(X\) be a smooth curve over the field of fractions of a complete discrete valuation ring and let \(\mathfrak{X}\) be a strongly semistable regular model of \(X\). It is possible to specialize a divisor on the curve to a divisor on the dual graph of the special fiber of \(\mathfrak{X}\); through this process the rank of the divisor can only increase.
The complete graph \(K_d\) pops up if we take a model of a smooth plane curve of degree d degenerating to a union of d lines. Moreover omitting edges from \(K_d\) can be interpreted as resolving singularities of a plane curve. In this talk we present some results on linear systems on complete graphs and complete graphs with a small number of omitted edges, and we will compare them with the corresponding results on plane curves. In particular, we compute the gonality sequence of complete graphs and the gonality of graphs obtained by omitting edges. We explain how to lift these graphs to curves with the same gonality using models of plane curves with nodes and harmonic morphisms. This is partially a joint work with Filip Cools.

Fei Xue (BMS)

19.10.16

16:30

MA 406

On Lattice Coverings by Simplices

Abstract:

By studying the volume of a generalized difference body, this talk presents the first nontrivial lower bound for the lattice covering density by \(n\)-dimensional simplices.

Ngoc Tran

05.10.16

14:45

MA 621

Tropical geometry and mechanism design

Abstract:

How to ensure an outcome based on collective decisions is 'better for all' when a) everyone acts in their personal interest only, and b) we do not know the exact 'motives' of each person? Crudely, this is the central problem in mechanism design, a branch of economics. In this talk, I show how insights from tropical geometry can solve such problems.

Thomas Kahle

05.10.16

13:30

MA 621

The geometry of rank-one tensor completion

Abstract:

tba

André Wagner

28.09.16

16:30

MA 621

Veronesean almost binomial almost complete intersections

Abstract:

The second Veronese ideal I_n contains a natural complete intersection J_n generated by the principal 2-minors of a symmetric (n × n)-matrix. In this talk I show, how to determine the subintersections of the primary decomposition of J_n where one intersectand is omitted. These subintersections can be described via combinatorial method and yield interesting insights into binomial ideals. This talk is based on joint work with Thomas Kahle.

Laura Silverstein (TU Wien)

27.07.16

16:30

MA 406

Tensor Valuations on Lattice Polytopes

Abstract:

A classification of symmetric tensor valuations on lattice polytopes in \(\mathbb{R}^n\) that intertwine the special linear group over the integers is established for the cases in which the rank of the tensor is at most \(n\). The scalar-valued case was classified by Betke and Kneser where it was shown that the only such valuations are the coefficients of the Ehrhart polynomial. Extending this result, the coefficients of the discrete moment tensor form a basis for the symmetric tensor valuations.

Apostolos Giannopoulos (University of Athens)

20.07.16

16:30

MA 406

Inequalities about sections and projections of convex bodies

Abstract:

We discuss lower dimensional versions of the slicing problem and of the Busemann-Petty problem, both in the classical setting and in the generalized setting of arbitrary measures in place of volume. We introduce an alternative approach which is based on the generalized Blaschke-Petkantschin formula, on asymptotic estimates for the dual affine quermassintegrals and on some new Loomis-Whitney type inequalities in the spirit of the uniform cover inequality of Bollobas and Thomason.

Gerassimos Barbatis (University of Athens)

13.07.16

16:30

MA 406

On the Hardy constant of some non-convex planar domains

Abstract:

The Hardy constant of a simply connected domain \(\Omega\subset\mathbb{R}^2\) is the best constant for the inequality
\[ \int_\Omega |\nabla u|^2 dx \geq c\int_\Omega \frac{u^2}{\operatorname{dist}(x,\partial\Omega)^2} dx, \quad u\in C_c^\infty(\Omega). \]
After the work of Ancona where the universal lower bound 1/16 was obtained, there has been a substantial interest on computing or estimating the Hardy constant of planar domains. In this talk we determine the Hardy constant of an arbitrary quadrilateral as well as of some other planar domains (joint work with Achilles Tertikas).

Matthew Trager

15.06.16

16:30

MA 621

Geometric models for computer vision

Abstract:

The goal of computer vision is to recover real-world information, either semantic (e.g., object recognition) or geometric (3D reconstruction), from sets of photographs or videos. In a broad sense, my research aims at investigating the fundamental geometric models, both synthetic and analytic, that can be used to describe shapes, cameras, and image contours, in processes of visual inference. In this talk, I will present some properties of 'silhouettes' that are obtained as projections of objects in space. I will also discuss some ongoing work that recasts traditional multi-view vision in terms of the Grassmannian of lines in P^3. This is joint work with Jean-Charles Faugere, Xavier Goaoc, Martial Hebert, Jean Ponce, Mohab Safey El Din and Bernd Sturmfels.

Axel Flinth

18.05.16

16:30

MA 406

High Dimensional Geometry in Compressed Sensing

Abstract:

The main aim of compressed sensing is to recover a low-dimensional object (e.g. a sparse vector, a low-rank matrix, ...) from few linear measurements. Many algorithms for this task (e.g. Basis Pursuit, nuclear norm-minimization) consists of minimizing a structure-promoting function \(f\) over the set of possible solutions.

An elegant approach to finding recovery guarantees of such programs relies on high-dimensional geometry of convex cones. The techniques were developed relatively recently, but have quickly gained popularity. In the first half of this talk, this approach will be presented in relative high detail.

In the second half of the talk, we will discuss the connection between geometry and stability of compressed sensing algorithms. In particular, we will present a new geometric criterion on \(A\) implying stability both with respect to model inexactness of the signal as well as to additive noise in the measurements.

Amanda Cameron

04.05.16

16:30

MA 621

A lattice point counting generalisation of the Tutte polynomial

Abstract:

The Tutte polynomial for matroids is not directly applicable to
polymatroids. For instance, deletion-contraction properties do not hold. We
construct a polynomial for polymatroids which behaves similarly to the
Tutte polynomial of a matroid, and in fact contains the same information as
the Tutte polynomial when we restrict to matroids. This polynomial is
constructed using lattice point counts in the Minkowski sum of the base
polytope of a polymatroid and scaled copies of the standard simplex. We
also show that, in the matroid case, our polynomial has coefficients of
alternating sign, with a combinatorial interpretation closely tied to the
Dawson partition.

Paul Breiding

27.04.16

16:30

MA 621

The spectral Atheory of tensors

Abstract:

The spectral theory of tensors aims at generalizing the concept of eigenvalues and eigenvectors of matrices to higher dimensional arrays. In this talk I will focus on the definition of E-eigenvalues and Z-eigenvalues given by Qi. After giving the definition itself I will explain various applications, give the higher dimensional equivalent of the characteristic polynomial and talk about the number of eigenvalues of both real and complex tensors. The last part of the talk will be about an algorithm to compute eigenvalues of tensors using homotopy methods and the complexity of this algorithm.

Shiri Artstein-Avidan (Tel Aviv University)

27.04.16

11:00

MA 406

On Godbersen’s conjecture and related inequalities

Abstract:

-

Stefan Weltge (Otto-von-Guericke Universität Magdeburg)

20.04.16

16:30

MA 406

Tight bounds on discrete quantitative Helly numbers

Abstract:

Given a subset S of R^n, let c(S,k) be the smallest number t such that
whenever finitely many convex sets have exactly k common points in S,
there exist at most t of these sets that already have exactly k common
points in S. For S = Z^n, this number was introduced by Aliev et al.
[2014] who gave an explicit bound showing that c(Z^n,k) = O(k) holds for
every fixed n. Recently, Chestnut et al. [2015] improved this to
c(Z^n,k) = O(k (log log k)(log k)^{-1/3} ) and provided the lower bound
c(Z^n,k) = Omega(k^{(n-1)/(n+1)}).
We provide a combinatorial description of c(S,k) in terms of polytopes
with vertices in S and use it to improve the previously known bounds as
follows: We strengthen the bound of Aliev et al. [2014] by a constant
factor and extend it to general discrete sets S. We close the gap for
Z^n by showing that c(Z^n,k) = Theta(k^{(n-1)/(n+1)}) holds for every
fixed n. Finally, we determine the exact values of c(Z^n,k) for all k <= 4.

Anna Lena Birkmeyer

16.03.16

16:30

MA 621

Realizability of Tropical Hypersurfaces in Matroid Fans

Abstract:

In tropical geometry, an important aim is to understand which tropical varieties arise as tropicalizations of algebraic varieties. We investigate this question in a relative setting: Given a matroid fan F coming from a linear space W, we ask if a tropical hypersurface in F is the tropicalization of an algebraic subvariety of W. We present an algorithm able to decide for any tropical hypersurface in F if it is realizable in W. Moreover, we use this algorithmic approach to describe the structure of the realization space of a tropical hypersurface H in F, i.e. the space of algebraic subvarieties of W tropicalizing to H, and show that the space of all realizable tropical hypersurfaces in F is an abstract polyhedral set.

Martin Genzel

02.03.16

16:30

MA 406

Convex Recovery of Structured Signals from Non-Linear Observations

Abstract:

In this talk, we study the problem of estimating a structured signal \(x_0\in\mathbb{R}\) linear Gaussian observations. Supposing that \(x_0\) belongs to a certain convex subset \(K\subset\mathbb{R}\) we will see that an accurate recovery is possible as long as the number of observations exceeds the effective dimension of \(K\), which is a common measure for the complexity of signal classes. Interestingly, it will turn out that the (possibly unknown) non-linearity of our model affects the error rate only by a multiplicative constant. This achievement is based on recent works by Yaniv Plan and Roman Vershynin, who have suggested to treat the non-linearity rather as noise which perturbs a linear measurement process. Using the concept of restricted strong convexity, we show that their results for the generalized Lasso can be extended to a fairly large class of convex loss functions. This is especially appealing for practical applications, since in many real-world scenarios, adapted loss functions empirically perform better than the classical square loss. To this end, the presented results provide a unified and general framework for signal reconstruction in high dimensions, covering various challenges from the fields of compressed sensing, signal processing, and statistical learning.

Mihalis Kolountzakis (University of Crete)

03.02.16

16:30

MA 406

Periodicity for tilings and spectra

Abstract:

We will talk about periodicity (and structure, more generally) in the study of tilings by translation, where the tile is a set or a function in an Abelian group, and also in the study of spectra of sets (sets of characters which form an orthogonal basis for L^2 of the set). There are connections to harmonic analysis, number theory, combinatorics and computation, and these make this subject so fascinating. Starting from the Fuglede conjecture, now disproved in dimension at least 3, which would connect tilings with spectra, we will go over cases where periodicity always holds, cases where it is optional and cases where it's never true, in one dimension (most positive results) and higher dimension (most interesting questions).

Simon Hampe

20.01.16

16:30

MA 621

Matroids over hyperfields

Abstract:

This talk is a summary of the very recent paper by Matthew Baker of the same title (http://arxiv.org/pdf/1601.01204.pdf). Various flavours of ''matroids with additional structure'', such as valuated and oriented matroids have been around for quite a while, with often very little connection between the different matroid species. Matroids over hyperfields are a generalization of all those concepts - thus allowing a unified treatment. In this talk I will first give a short introduction to hyperfields. I will then define what matroids over hyperfields are, give examples and discuss generalizations of familiar matroid operations such as duality and minors.

Silouanos Brazitikos (University of Athens)

13.01.16

16:30

MA 406

Quantitative versions of Helly's theorem

Abstract:

We provide a new quantitative version of Helly's theorem: there exists an absolute constant\(\alpha >1\)with the following property: if\(\{P_i: i\in I\}\)is a finite family of convex bodies in\({\mathbb R}^n\)with\({\rm int}\left (\bigcap_{i\in I}P_i\right)\neq\emptyset\), then there exist\(z\in {\mathbb R}^n\),\(s\ls \alpha n\)and\(i_1,\ldots i_s\in I\)such that \begin{equation*} z+P_{i_1}\cap\cdots\cap P_{i_s}\subseteq cn^{3/2}\left(z+\bigcap_{i\in I}P_i\right), \end{equation*} where\(c>0\)is an absolute constant. This directly gives a version of the 'quantitative' volume and diameter theorem of
> B\'{a}r\'{a}ny, Katchalski and Pach, with a polynomial dependence on the dimension.

Kristin Shaw

06.01.16

16:30

MA 621

What is Étale cohomology?

Abstract:

Wished for by Weil and defined by Grothendieck, Étale cohomology is to varieties over fields of finite characteristic what usual cohomology is to complex or real algebraic varieties when they are equipped with the analytic, respectively Euclidean topology.
This very short introduction will motivate and try to explain the basic ingredients of Étale cohomology. Throughout I will make analogies to the complex and real situations and stick to simple examples in hopes to provide some intuition into this deep theory.

Benjamin Schröter

02.12.15

16:30

MA 621

The degree of a tropical basis

Abstract:

In this talk I will present my work with Michael Joswig. I will give an explicit bound on the degree of a tropical basis. Our result is derived from the algorithm of Bogart, Jensen, Speyer, Sturmfels and Thomas that computes a tropical variety via Groebner bases and saturation. Furthermore I will give examples that illustrate the difference between tropical and Groebner bases.

Bernardo González Merino (TU München)

18.11.15

16:30

MA 406

On the Minkowski measure of symmetry

Abstract:

The Minkowski measure of symmetry s(K) of a convex body K, is the smallest positive dilatation of K containing a translate of -K. In this talk we will explain some of its basic properties in detail.

Afterwards, we will show how s(.) can be used to strengthen, smoothen, and join different geometric inequalities, as well as its connections to other concepts such as diametrical compleness, Jung's inequality, or Banach-Mazur distance.

André Wagner

04.11.15

16:30

MA 621

Introduction to singular value decomposition

Abstract:

In this short talk I will give a brief introduction to singular value decomposition, some of it's applications and efficient ways to compute it.

Jonathan Spreer

28.10.15

16:30

MA 621

Algorithms and complexity for Turaev-Viro invariants

Abstract:

The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. I will discuss this family of invariants, and present an explicit fixed-parameter tractable algorithm for arbitrary r which is practical -- and indeed preferable -- to the prior state of the art for real computation. This is joint work with Ben Burton and Clément Maria

Alexander Gamkrelidze

11.08.15

15:00

MA 621

Algorithms for Convex Topology: Computation of Hom-Complexes

Abstract:

After a brief introduction to the field of polytopal complexes and affine maps between them, we define the set Hom(P,Q) of all possible affine mappings between two polytopal complexes P and Q that builds a polytopal complex in a higher dimensional space. There are efficient software packages like Polymake that deal (among other things) with the computation of Hom-complexes between two polytopes, but no complexes. In this talk, we introduce algorithms that compute the above mentioned structures and their homotopies. At the end of the talk we introduce a conjectured geometric analog to a well known formula Hom(P,ker(f))=kerHom(P,f): Hom(P,\(\Sigma\)(f))=\(\Sigma\) Hom(P,f) and discuss some ideas to compute the Sigma operator and the possible ways to prove the equation. In case the above conjecture holds, the wide area of the algebraic theory of polytopes can be established.